Each n-by-n matrix with n > 1 is a sum of 5 coninvolutory matrices

An n×nn×n complex matrix A is called coninvolutory   if A¯A=In and skew-coninvolutory   if A¯A=−In (which implies that n   is even). We prove that each matrix of size n×nn×n with n>1n>1 is a sum of 5 coninvolutory matrices and each matrix of size 2m×2m2m×2m is a sum of 5 skew-coninvolutory matrices.We also prove that each square complex matrix is a sum of a coninvolutory matrix and a condiagonalizable matrix. A matrix M is called condiagonalizable   if M=S¯−1DS in which S is nonsingular and D is diagonal.